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<HTML>
<HEAD><TITLE>AB13AX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB13AX">AB13AX</A></H2>
<H3>
Hankel-norm of a stable system with the state dynamics matrix in real Schur form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute the Hankel-norm of the transfer-function matrix G of
a stable state-space system (A,B,C). The state dynamics matrix A
of the given system is an upper quasi-triangular matrix in
real Schur form.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
DOUBLE PRECISION FUNCTION AB13AX( DICO, N, M, P, A, LDA, B, LDB,
$ C, LDC, HSV, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
</PRE>
<B><FONT SIZE="+1">Function Value</FONT></B>
<PRE>
AB13AX DOUBLE PRECISION
The Hankel-norm of G (if INFO = 0).
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
DICO CHARACTER*1
Specifies the type of the system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the state-space representation, i.e. the
order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A in a real Schur canonical form.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, this array contains the Hankel singular
values of the given system ordered decreasingly.
HSV(1) is the Hankel norm of the given system.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system, and let G be the corresponding
transfer-function matrix. The Hankel-norm of G is computed as the
the maximum Hankel singular value of the system (A,B,C).
The computation of the Hankel singular values is performed
by using the square-root method of [1].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method relies on a square-root technique.
3
The algorithms require about 17N floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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